Self-adjoint Boundary Conditions Research Articles

Dependence of eigenvalues of fourth-order Sturm-Liouville problems on canonical boundary conditions

In this paper, we discuss the dependence of eigenvalues of fourth-order Sturm-Liouville (SL) problems on the fundamental canonical forms of separated, coupled and mixed self-adjoint boundary conditions. The eigenvalues depend not only continuously but smoothly on the boundary conditions and boundary points. Simultaneously, the derivative expression of the eigenvalues with respect to all the parameters in the boundary conditions are given, and thus its monotonicity with respect to some parameters is derived.

Characterization of self-adjoint domains for regular odd order C-symmetric differential operators

We enlarge the class of regular odd order differential operators and find the self-adjoint boundary condition of every operator. In this paper, we give the characterization of self-adjoint domains for regular odd order C-symmetric differential operators with two-point boundary conditions. The previously known characterization of self-adjoint operators for regular odd order symmetric differential operators is the special case of this one.

Bogoliubov-de Gennes equation on graphs: A model for tree-branched Majorana wire network

We consider Bogoliubov-de Gennes equation on a metric tree graph. Formulation of the problem for arbitrary graph topology is provided. Self-adjoint vertex boundary conditions are derived. Exact solutions of the problem is obtained for quantum tree graph. A quantum graph based model for tree-branched Majorana wire network is proposed.

A 4D IIB flux vacuum and supersymmetry breaking. Part II. Bosonic spectrum and stability

We recently constructed type-IIB compactifications to four dimensions depending on a single additional coordinate, where a five-form flux Φ on an internal torus leads to a constant string coupling. Supersymmetry is fully broken when the internal manifold includes a finite interval of length ℓ, which is spanned by a conformal coordinate in a finite range 0 < z < zm. Here we examine the low-lying bosonic spectra and their classical stability, paying special attention to self-adjoint boundary conditions. Special boundary conditions result in the emergence of zero modes, which are determined exactly by first-order equations. The different sectors of the spectrum can be related to Schrödinger operators on a finite interval, characterized by pairs of real constants μ and μ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{\\mu } $$\\end{document}, with μ equal to 1/3 or 2/3 in all cases and different values of μ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{\\mu } $$\\end{document}. The potentials behave as μ2−1/4z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{\\mu^2-1/4}{z^2} $$\\end{document} and μ~2−1/4zm−z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{{\\overset{\\sim }{\\mu}}^2-1/4}{{\\left({z}_m-z\\right)}^2} $$\\end{document} near the ends and can be closely approximated by exactly solvable trigonometric ones. With vanishing internal momenta, one can thus identify a wide range of boundary conditions granting perturbative stability, despite the intricacies that emerge in some sectors. For the Kaluza-Klein excitations of non-singlet vectors and scalars the Schrödinger systems couple pairs of fields, and the stability regions, which depend on the background, widen as the ratio Φ/ℓ4 decreases.

Classification of self-adjoint domains of odd-order differential operators with matrix theory

Abstract In this article, we investigate the classification of self-adjoint boundary conditions of odd-order differential operators. We obtain that for odd-order self-adjoint boundary conditions under some assumptions, there are exactly two basic types of self-adjoint boundary conditions: coupled and mixed. Moreover we determine the number of possible conditions for each type, which is different from the even-order cases. Our construction will play an important role in the canonical forms and in the spectral analysis of these operators.

Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions

We investigate the controllability of an infinite-dimensional quantum system: a quantum particle confined on a Thick Quantum Graph, a generalisation of Quantum Graphs whose edges are allowed to be manifolds of arbitrary dimension with quasi-δ boundary conditions. This is a particular class of self-adjoint boundary conditions compatible with the graph structure. We prove that global approximate controllability can be achieved using two physically distinct protocols: either using the boundary conditions as controls, or using time-dependent magnetic fields. Both cases have time-dependent domains for the Hamiltonians.

General pseudo self-adjoint boundary conditions for a 1D KFG particle in a box

We consider a 1D Klein–Fock–Gordon particle in a finite interval, or box. We construct for the first time the most general set of pseudo self-adjoint boundary conditions for the Hamiltonian operator that is present in the first order in time 1D Klein–Fock–Gordon wave equation, or the 1D Feshbach–Villars wave equation. We show that this set depends on four real parameters and can be written in terms of the one-component wavefunction for the second order in time 1D Klein–Fock–Gordon wave equation and its spatial derivative, both evaluated at the endpoints of the box. Certainly, we write the general set of pseudo self-adjoint boundary conditions also in terms of the two-component wavefunction for the 1D Feshbach–Villars wave equation and its spatial derivative, evaluated at the ends of the box; however, the set actually depends on these two column vectors each multiplied by the singular matrix that is present in the kinetic energy term of the Hamiltonian. As a consequence, we found that the two-component wavefunction for the 1D Feshbach–Villars equation and its spatial derivative do not necessarily satisfy the same boundary condition that these quantities satisfy when multiplied by the singular matrix. In any case, given a particular boundary condition for the one-component wavefunction of the standard 1D Klein–Fock–Gordon equation and using the pair of relations that arise from the very definition of the two-component wavefunction for the 1D Feshbach–Villars equation, the respective boundary condition for the latter wavefunction and its derivative can be obtained. Our results can be extended to the problem of a 1D Klein–Fock–Gordon particle moving on a real line with a point interaction (or a hole) at one point.

Dirac field in AdS2 and representations of SL̃(2,R)

We analyze a massive spinor field satisfying the Dirac equation in the universal covering space of two-dimensional anti-de Sitter space. In order to obtain well-defined dynamics for the classical field despite the lack of global-hyperbolicity of the spacetime, we impose a suitable set of boundary conditions that render the spatial component of the Dirac operator self-adjoint. Then, we find which of the solution spaces obtained by imposing the self-adjoint boundary conditions are invariant under the action of the isometry group of the spacetime manifold, namely, the universal covering group of SL(2,R). The invariant solution spaces are then identified with unitary irreducible representations of this group using the classification given by Pukánszky [Math. Ann. 156, 96–143 (1964)]. We determine which of these correspond to invariant positive- or negative-frequency subspaces and, thus, result in a vacuum state invariant under the isometry group after canonical quantization. Additionally, we examine the invariant theories obtained from the self-adjoint boundary conditions, which result in a non-invariant vacuum state, identifying the unitary representation this state belongs to.

The matrix nonlinear Schrödinger equation with a potential

This paper is devoted to the study of the large-time asymptotics of the small solutions to the matrix nonlinear Schrödinger equation with a potential on the half-line and with general selfadjoint boundary condition, and on the line with a potential and a general point interaction, in the whole supercritical regime. We prove that the small solutions are scattering solutions that asymptotically in time, t→±∞, behave as solutions to the associated linear matrix Schrödinger equation with the potential identically zero. The potential can be either generic or exceptional. Our approach is based on detailed results on the spectral and scattering theory for the associated linear matrix Schrödinger equation with a potential, and in a factorization technique that allows us to control the large-time behavior of the solutions in appropriate norms.

Left-definite Hamiltonian systems and corresponding nested circles

This work aims to construct the Titchmarsh-Weyl $M(\lambda )-$theory for an even-dimensional left-definite Hamiltonian system. For this purpose, we introduce a suitable Lagrange formula and selfadjoint boundary conditions including the spectral parameter $\lambda $. Then we obtain circle equations having nesting properties. Using the intersection point belonging to all the circles we share a lower bound for the number of Dirichlet-integrable solutions of the system.

Scalar field in AdS2 and representations of SL̃(2,R)

We study the solutions to the Klein–Gordon equation for the massive scalar field in the universal covering space of a two-dimensional anti-de Sitter space. For certain values of the mass parameter, we impose a suitable set of boundary conditions, which make the spatial component of the Klein–Gordon operator self-adjoint. This makes the time-evolution of the classical field well defined. Then, we use the transformation properties of the scalar field under the isometry group of the theory, namely, the universal covering group of SL(2,R), in order to determine which self-adjoint boundary conditions are invariant under this group and which lead to the positive-frequency solutions forming a unitary representation of this group and, hence, to a vacuum state invariant under this group. Then, we examine the cases where the boundary condition leads to an invariant theory with a non-invariant vacuum state and determine the unitary representation to which the vacuum state belongs.

Wave and scattering operators for the nonlinear matrix Schrödinger equation on the half-line with a potential

We consider the nonlinear matrix Schrödinger equation on the half-line with general selfadjoint boundary condition. We prove the existence of local solutions in the corresponding energy space. Moreover, we consider the scattering problem for this problem in the scattering-supercritical regime for both generic and exceptional potentials. We construct the wave, inverse wave and scattering operators in a certain neighborhood of the origin in the scattering space Σ=H1∩L2,1 for generic potentialsin L1,2+η, with η>1/2, and for exceptional potentials in L1,3 under further assumptions on the input data. In particular, thanks to the general boundary conditions, we are able to obtain a scattering result for the nonlinear matrix Schrödinger equation on the line with a potential and point interactions.

The $L^p$ boundedness of the wave operators for matrix Schrödinger equations

We prove that the wave operators for n \times n matrix Schrödinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces L^p(\mathbb R^+, \mathbb C^n), 1 &lt; p &lt; \infty, for slowly decaying selfadjoint matrix potentials V that satisfy the condition \int_{0}^{\infty}(1+x) |V(x)|\: dx &lt; \infty. Moreover, assuming that \int_{0}^{\infty }(1+x^\gamma) |V(x)|\: dx &lt; \infty, \gamma &gt; \frac{5}{2}, and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in L^1(\mathbb R^+, \mathbb C^n) and in L^\infty(\mathbb R^+, \mathbb C^n). We also prove that the wave operators for n\times n matrix Schrödinger equations on the line are bounded in the spaces L^p(\mathbb R, \mathbb C^n), 1 &lt; p &lt; \infty, assuming that the perturbation consists of a point interaction at the origin and of a potential \mathcal V that satisfies the condition \int_{-\infty}^{\infty}(1+|x|)|\mathcal V(x)|\: dx &lt; \infty. Further, assuming that \int_{-\infty}^{\infty }(1+|x|^\gamma) |\mathcal V(x)|\: dx &lt; \infty, \gamma &gt; \frac{5}{2}, and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in L^1(\mathbb R, \mathbb C^n) and in L^\infty(\mathbb R, \mathbb C^n). We obtain our results for n\times n matrix Schrödinger equations on the line from the results for 2n\times 2n matrix Schrödinger equations on the half line.

Anderson Localisation for Quasi-One-Dimensional Random Operators

In 1990, Klein, Lacroix, and Speis proved (spectral) Anderson localisation for the Anderson model on the strip of width W geqslant 1, allowing for singular distribution of the potential. Their proof employs multi-scale analysis, in addition to arguments from the theory of random matrix products (the case of regular distributions was handled earlier in the works of Goldsheid and Lacroix by other means). We give a proof of their result avoiding multi-scale analysis and also extend it to the general quasi-one-dimensional model, allowing, in particular, random hopping. Furthermore, we prove a sharp bound on the eigenfunction correlator of the model, which implies exponential dynamical localisation and exponential decay of the Fermi projection. The method is also applicable to operators on the half-line with arbitrary (deterministic, self-adjoint) boundary condition. Our work generalises and complements the single-scale proofs of localisation in pure one dimension (W=1), recently found by Bucaj–Damanik–Fillman–Gerbuz–VandenBoom–Wang–Zhang, Jitomirskaya–Zhu, Gorodetski–Kleptsyn, and Rangamani.

Spectral measures for derivative powers via matrix-valued Clark theory

The theory of finite-rank perturbations allows for the determination of spectral information for broad classes of operators using the tools of analytic function theory. In this work, finite-rank perturbations are applied to powers of the derivative operator, providing a full account from self-adjoint boundary conditions to computing aspects of the operators' matrix-valued spectral measures. In particular, the support and weights of the Clark (spectral) measures are computed via the connection between matrix-valued contractive analytic functions and matrix-valued nonnegative measures through the Herglotz Representation Theorem. For operators associated with several powers of the derivative, explicit formulae for these measures are included. While eigenfunctions and eigenvalues for these operators with fixed boundary conditions can often be computed using direct methods from ordinary differential equations, this approach provides a more complete picture of the spectral information.

Inverse problem solution and spectral data characterization for the matrix Sturm–Liouville operator with singular potential

The self-adjoint matrix Sturm–Liouville operator on a finite interval with singular potential of class $$W_2^{-1}$$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm–Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for the studied operator. In addition, the main results are applied to the Sturm–Liouville operator on a graph of arbitrary geometrical structure.

Dirac particles on periodic quantum graphs.

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasiperiodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum graphs is obtained. Band spectra of the periodic quantum graphs of different topologies are calculated. Universality of the probability to be in the spectrum for certain graph topologies is observed.

Direct and Inverse Problems for the Matrix Sturm–Liouville Operator with General Self-Adjoint Boundary Conditions

The matrix Sturm–Liouville operator on a finite interval with boundary conditions in general self-adjoint form and with singular potential of class $$W_2^{-1}$$ is studied. This operator generalizes Sturm–Liouville operators on geometrical graphs. We investigate structural and asymptotical properties of the spectral data (eigenvalues and weight matrices) of this operator. Furthermore, we prove the uniqueness of recovering the operator from its spectral data, by using the method of spectral mappings.

Coupling constant dependence for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schr\"odinger equation $-f''+q_\alpha f = Ef$ on the positive half-axis with the potential $q_\alpha(r)=(\alpha-1/4)r^{-2}$. It is known that the value $\alpha=0$ plays a special role in this problem: all self-adjoint realizations of the formal differential expression $-\partial^2_r + q_\alpha(r)$ for the Hamiltonian have infinitely many eigenvalues for $\alpha<0$ and at most one eigenvalue for $\alpha\geq 0$. We find a parametrization of self-adjoint boundary conditions and eigenfunction expansions that is analytic in $\alpha$ and, in particular, is not singular at $\alpha = 0$. Employing suitable singular Titchmarsh--Weyl $m$-functions, we explicitly find the spectral measures for all self-adjoint Hamiltonians and prove their smooth dependence on $\alpha$ and the boundary condition. Using the formulas for the spectral measures, we analyse in detail how the "phase transition" through the point $\alpha=0$ occurs for both the eigenvalues and the continuous spectrum of the Hamiltonians.

The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.